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  <div class="section" id="numpy-linalg-pinv">
<h1>numpy.linalg.pinv<a class="headerlink" href="#numpy-linalg-pinv" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.linalg.pinv">
<code class="sig-prename descclassname">numpy.linalg.</code><code class="sig-name descname">pinv</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">rcond=1e-15</em>, <em class="sig-param">hermitian=False</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/linalg/linalg.py#L1881-L1970"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.linalg.pinv" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the (Moore-Penrose) pseudo-inverse of a matrix.</p>
<p>Calculate the generalized inverse of a matrix using its
singular-value decomposition (SVD) and including all
<em>large</em> singular values.</p>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 1.14: </span>Can now operate on stacks of matrices</p>
</div>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>a</strong><span class="classifier">(…, M, N) array_like</span></dt><dd><p>Matrix or stack of matrices to be pseudo-inverted.</p>
</dd>
<dt><strong>rcond</strong><span class="classifier">(…) array_like of float</span></dt><dd><p>Cutoff for small singular values.
Singular values less than or equal to
<code class="docutils literal notranslate"><span class="pre">rcond</span> <span class="pre">*</span> <span class="pre">largest_singular_value</span></code> are set to zero.
Broadcasts against the stack of matrices.</p>
</dd>
<dt><strong>hermitian</strong><span class="classifier">bool, optional</span></dt><dd><p>If True, <em class="xref py py-obj">a</em> is assumed to be Hermitian (symmetric if real-valued),
enabling a more efficient method for finding singular values.
Defaults to False.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.17.0.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>B</strong><span class="classifier">(…, N, M) ndarray</span></dt><dd><p>The pseudo-inverse of <em class="xref py py-obj">a</em>. If <em class="xref py py-obj">a</em> is a <em class="xref py py-obj">matrix</em> instance, then so
is <em class="xref py py-obj">B</em>.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>LinAlgError</strong></dt><dd><p>If the SVD computation does not converge.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The pseudo-inverse of a matrix A, denoted <img class="math" src="../../_images/math/56f8c7fcc1750e00c006d70b8e0803cfcad72d06.svg" alt="A^+"/>, is
defined as: “the matrix that ‘solves’ [the least-squares problem]
<img class="math" src="../../_images/math/6525336972790f2bd6fb9c5d9ed3a17262d22fe9.svg" alt="Ax = b"/>,” i.e., if <img class="math" src="../../_images/math/9c3bdbab48a2177b8e46ec59561cee6f11e3ed28.svg" alt="\bar{x}"/> is said solution, then
<img class="math" src="../../_images/math/56f8c7fcc1750e00c006d70b8e0803cfcad72d06.svg" alt="A^+"/> is that matrix such that <img class="math" src="../../_images/math/0e4460898f3785485292ad085480da1cce832e7c.svg" alt="\bar{x} = A^+b"/>.</p>
<p>It can be shown that if <img class="math" src="../../_images/math/0d63aaca9e9a497e49ca920edba5c248d789c4d5.svg" alt="Q_1 \Sigma Q_2^T = A"/> is the singular
value decomposition of A, then
<img class="math" src="../../_images/math/0357ab919256ac3697acdd09a08a66b5ea6e5f8f.svg" alt="A^+ = Q_2 \Sigma^+ Q_1^T"/>, where <img class="math" src="../../_images/math/ff60f407f4f25b34a6e6242f49ebf83c5319fd11.svg" alt="Q_{1,2}"/> are
orthogonal matrices, <img class="math" src="../../_images/math/6edc5c119344e25a06e6ac4cb56f2d5e2f09a2f1.svg" alt="\Sigma"/> is a diagonal matrix consisting
of A’s so-called singular values, (followed, typically, by
zeros), and then <img class="math" src="../../_images/math/b90108759cd3dca6ed5afff7d55805bd675008d0.svg" alt="\Sigma^+"/> is simply the diagonal matrix
consisting of the reciprocals of A’s singular values
(again, followed by zeros). <a class="reference internal" href="#rec505eafac9d-1" id="id1">[1]</a></p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="rec505eafac9d-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>G. Strang, <em>Linear Algebra and Its Applications</em>, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pp. 139-142.</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>The following example checks that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">*</span> <span class="pre">a+</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">==</span> <span class="pre">a</span></code> and
<code class="docutils literal notranslate"><span class="pre">a+</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">a+</span> <span class="pre">==</span> <span class="pre">a+</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">a</span><span class="p">)))</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">B</span><span class="p">)))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

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